Solve for x log of x+ log of x+48=2

Problem

(log_)(x)+(log_)(x+48)=2

Solution

1. Apply the product rule for logarithms, which states that (log_)(a)+(log_)(b)=(log_)(a*b)

(log_)(x*(x+48))=2

2. Rewrite in exponential form by using the base of the common logarithm, which is 10

x*(x+48)=10

3. Expand and simplify the equation to form a standard quadratic equation.

x2+48*x=100

x2+48*x−100=0

4. Factor the quadratic by finding two numbers that multiply to −100 and add to 48

(x+50)*(x−2)=0

5. Solve for x by setting each factor equal to zero.

x=−50

x=2

6. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since (log_)(−50) is undefined, we discard x=−50

x=2

Final Answer

(log_)(x)+(log_)(x+48)=2⇒x=2

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